Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 39 / 3 / ii

For a stopping time $\tau$, $U_{t\wedge \tau }$ is adapted: split according to $\{\tau =j\}$ for $j<t$ and $\{\tau \ge t\}$, all events belonging to ${\mathcal{F}}_t$. It is integrable because $|U_{t\wedge \tau }|\le {\sum }_{j=0}^t|U_j|$. Moreover,The indicator is ${\mathcal{F}}_t$-measurable, so conditioning and the supermartingale property giveThus $\boxed{(U_{t\wedge \tau }{)}_t\text{ is a supermartingale}}$. This proves stopping preserves supermartingales in discrete time without assuming $\tau$ itself bounded or integrable.